mixed base Heine-type transformations

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21: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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22: 18.17 Integrals

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§18.17(v) Fourier Transforms

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Jacobi

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Ultraspherical

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§18.17(vi) Laplace Transforms

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§18.17(vii) Mellin Transforms

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23: 2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of $f(t)$ is defined by … ► $ℳf\left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). … ►Corresponding results for the generalized Stieltjes transform … ►where $ℳf\left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. …

24: 17.12 Bailey Pairs

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Bailey Transform

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17.12.3 $${\beta }_n=\sum _{j=0}^n\frac{{\alpha }_j}{{(q;q)}_{n-j}{(aq;q)}_{n+j}}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

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17.12.4 $$\sum _{n=0}^{\mathrm{\infty }}{q}^{n^2}{a}^n{\beta }_n=\frac{1}{{(aq;q)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}{q}^{n^2}{a}^n{\alpha }_n.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

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25: 2.3 Integrals of a Real Variable

… ►Assume that the Laplace transform …Then … ►where $ℳf\left(\alpha \right)$ is the Mellin transform of $f(t)$ (§2.5(i)). … ►The integral (2.3.24) transforms into … ►

§2.3(vi) Asymptotics of Mellin Transforms

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26: Guide to Searching the DLMF

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Table 1: Query Examples

► ►►►►►

Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
…
Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform"	both $1/(2\pi )$ and the phrase “Fourier transform”.
…

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… ►Sometimes there are distinctions between various special function names based on font style, such as the use of bold or calligraphic letters. …

27: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

28: 17.11 Transformations of $q$-Appell Functions

§17.11 Transformations of $q$-Appell Functions

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17.11.1 $${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)=\frac{{(a,bx,b^{\prime }y;q)}_{\mathrm{\infty }}}{{(c,x,y;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c/a,x,y}{bx,b^{\prime }y};q,a),$$

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Symbols:

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $x$: real variable and $y$: real variable

Referenced by:

§17.11, Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.11.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

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17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.11.2)

Permalink:

http://dlmf.nist.gov/17.11.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

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17.11.3 $${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)=\frac{{(a,bx;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a^{\prime },b^{\prime };q)}_n{(x;q)}_r{(c/a;q)}_{n+r}{a}^r{y}^n}{{(q,c/a;q)}_n{(q,bx;q)}_r}.$$

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Symbols:

${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$: third $q$-Appell function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§17.11, Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.11.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

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17.11.4 $$\sum _{m_1,\mathrm{\dots },m_n\geqq 0}\frac{{(a;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}{(b_1;q)}_{m_1}{(b_2;q)}_{m_2}\mathrm{\cdots }{(b_n;q)}_{m_n}{x}_1^{m_1}{x}_2^{m_2}\mathrm{\cdots }{x}_n^{m_n}}{{(q;q)}_{m_1}{(q;q)}_{m_2}\mathrm{\cdots }{(q;q)}_{m_n}{(c;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}}=\frac{{(a,b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n;q)}_{\mathrm{\infty }}}{{(c,x_1,x_2,\mathrm{\dots },x_n;q)}_{\mathrm{\infty }}}{}_{n+1}{}^{}\varphi _n^{}(\genfrac{}{}{0pt}{}{c/a,x_1,x_2,\mathrm{\dots },x_n}{b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n};q,a).$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.11 and Ch.17

29: 23.15 Definitions

… ►Also $𝒜$ denotes a bilinear transformation on $\tau$, given by ►

23.15.3 $$𝒜\tau =\frac{a\tau +b}{c\tau +d},$$

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Symbols:

$\tau$: complex variable, $𝒜$: bilinear transformation, $a$: integer, $b$: integer, $c$: integer and $d$: integer

Referenced by:

§20.11(i), §23.18

Permalink:

http://dlmf.nist.gov/23.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

… ►The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … ►

23.15.9 $$\eta \left(\tau \right)={\left(\frac{1}{2}{\theta }_1^{\prime }(0,q)\right)}^{1/3}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}{\theta }_3\left(\frac{1}{2}\pi (1+\tau )\right|3\tau ).$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), ${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nome and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.15(ii), §23.17(iii)

Permalink:

http://dlmf.nist.gov/23.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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30: 8.19 Generalized Exponential Integral

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8.19.2 $$E_p\left(z\right)=z^{p-1}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t^p}dt.$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§8.19(i), §8.19(viii)

Permalink:

http://dlmf.nist.gov/8.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19 and Ch.8

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8.19.3 $$E_p\left(z\right)={\int }_1^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^p}dt,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

A&S Ref:

5.1.4

Referenced by:

§8.19(iii)

Permalink:

http://dlmf.nist.gov/8.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

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8.19.4 $$E_p\left(z\right)=\frac{z^{p-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{p-1}{\mathrm{e}}^{-zt}}{1+t}dt,$$ , $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/8.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

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8.19.5 $$E_0\left(z\right)=z^{-1}{\mathrm{e}}^{-z},$$ $z\ne 0$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $z$: complex variable

A&S Ref:

5.1.24

Permalink:

http://dlmf.nist.gov/8.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iii), §8.19 and Ch.8

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8.19.12 $$p{E}_{p+1}\left(z\right)+z{E}_p\left(z\right)={\mathrm{e}}^{-z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19 and Ch.8

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